A new dark sector antibaryon, denoted ψ_{D}, could be produced in decays of B mesons. This Letter presents a search for B^{+}→ψ_{D}+p (and the charge conjugate) decays in e^{+}e^{-} annihilations at 10.58 GeV, using data collected in the BABAR experiment. Data corresponding to an integrated luminosity of 398 fb^{-1} are analyzed. No evidence for a signal is observed. Branching fraction upper limits in the range from 10^{-7}-10^{-5} are obtained at 90% confidence level for masses of 1.0<m_{ψ_{D}}<4.3 GeV/c^{2}. The result is also reinterpreted to provide the first limits on a supersymmetric model with R-parity violation and a light neutralino.